The Normal Distribution

A normal distribution is symmetrical and bell-shaped, written $N(\mu, \sigma^2)$ where $\mu$ is the mean and $\sigma$ is the standard deviation. As a rule of thumb: about 68\% of data lies within 1 standard deviation of the mean, about 95\% within 2 s.d., and virtually all of it within 3 s.d.

Many real-world measurements — heights, reaction times, manufacturing errors — are well modelled by a normal distribution, even though it's always an approximation to the real data.

To find a probability for any normal distribution, you first convert ("standardise") a value $x$ into a z-value, which tells you how many standard deviations $x$ is from the mean: $$z = \frac{x-\mu}{\sigma}$$ This converts $N(\mu,\sigma^2)$ into the standard normal distribution $N(0,1)$, for which probabilities are tabulated as $\Phi(z) = P(Z

Use your calculator or statistical tables to look up $\Phi(z)$ for a given z-value, or to work the other way: given a probability, find the z-value (and hence the original value of $x$) using the inverse normal function. Always sketch the curve and shade the region you want — it makes it far easier to spot whether you need $\Phi(z)$, $1-\Phi(z)$, or a difference of two areas.